Preheat the oven to 400 degrees and lightly grease a cast iron skillet.
Mix together the cream and milk and set aside.
In the skillet, arrange the potatoes in an even layer. Drizzle with a few tablespoons of the cream mixture and season with salt, pepper, thyme and dust with some of the cheeses. Repeat with the remaining potatoes and cream, seasoning with salt and pepper, to form 6-8 layers. Press down on the layers to totally submerge the sweet potatoes in the cream mixture and top with additional cheese if needed.
Cover and bake for 30 minutes. Remove the cover and continue baking until the cream has been absorbed, the potatoes are cooked through, and the top is browned, 30 to 45 minutes. Remove from the oven and let rest for 15 minutes before serving.